In a previous work, the authors introduced the class of "graphs with bounded induced distance of order k", (BID(k) for short) to model non-reliable interconnection networks. A network modeled as a graph in BID(k) can be characterized as follows: if some nodes have failed, as long as two nodes remain connected, the distance between these nodes in the faulty graph is at most k times the distance in the non-faulty graph. The smallest k such that G is in BID(k) is called "stretch number" of G. In this paper we give new characterization, algorithmic, and existence results about graphs with small stretch number.
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